# casinhf, casinh, casinhl

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< c‎ | numeric‎ | complex

Defined in header ``<complex.h>`` |  |
float complex casinhf( float complex z ); | (1) | (since C99)
double complex casinh( double complex z ); | (2) | (since C99)
long double complex casinhl( long double complex z ); | (3) | (since C99)
Defined in header ``<tgmath.h>`` |  |
#define asinh( z ) | (4) | (since C99)

1-3) Computes the complex arc hyperbolic sine of `z` with branch cuts outside the interval [−i; +i] along the imaginary axis.
4) Type-generic macro: If `z` has type long double complex, `casinhl` is called. if `z` has type double complex, `casinh` is called, if `z` has type float complex, `casinhf` is called. If `z` is real or integer, then the macro invokes the corresponding real function (asinhf, asinh, asinhl). If `z` is imaginary, then the macro invokes the corresponding real version of the function asin, implementing the formula asinh(iy) = i asin(y), and the return type is imaginary.

### Parameters

z | - | complex argument

### Return value

If no errors occur, the complex arc hyperbolic sine of `z` is returned, in the range of a strip mathematically unbounded along the real axis and in the interval [−iπ/2; +iπ/2] along the imaginary axis.

### Error handling and special values

Errors are reported consistent with math_errhandling

If the implementation supports IEEE floating-point arithmetic,

- casinh(conj(z)) == conj(casinh(z))
- casinh(-z) == -casinh(z)
- If `z` is `+0+0i`, the result is `+0+0i`
- If `z` is `x+∞i` (for any positive finite x), the result is `+∞+π/2`
- If `z` is `x+NaNi` (for any finite x), the result is `NaN+NaNi` and FE_INVALID may be raised
- If `z` is `+∞+yi` (for any positive finite y), the result is `+∞+0i`
- If `z` is `+∞+∞i`, the result is `+∞+iπ/4`
- If `z` is `+∞+NaNi`, the result is `+∞+NaNi`
- If `z` is `NaN+0i`, the result is `NaN+0i`
- If `z` is `NaN+yi` (for any finite nonzero y), the result is `NaN+NaNi` and FE_INVALID may be raised
- If `z` is `NaN+∞i`, the result is `±∞+NaNi` (the sign of the real part is unspecified)
- If `z` is `NaN+NaNi`, the result is `NaN+NaNi`

### Notes

Although the C standard names this function "complex arc hyperbolic sine", the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse hyperbolic sine", and, less common, "complex area hyperbolic sine".

Inverse hyperbolic sine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segments (-i∞,-i) and (i,i∞) of the imaginary axis.

The mathematical definition of the principal value of the inverse hyperbolic sine is asinh z = ln(z + √1+z2
)

For any z, asinh(z) =

asin(iz)
i

### Example

Run this code

    #include <stdio.h>
    #include <complex.h>

    int main(void)
    {
        double complex z = casinh(0+2*I);
        printf("casinh(+0+2i) = %f%+fi\n", creal(z), cimag(z));

        double complex z2 = casinh(-conj(2*I)); // or casinh(CMPLX(-0.0, 2)) in C11
        printf("casinh(-0+2i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));

        // for any z, asinh(z) = asin(iz)/i
        double complex z3 = casinh(1+2*I);
        printf("casinh(1+2i) = %f%+fi\n", creal(z3), cimag(z3));
        double complex z4 = casin((1+2*I)*I)/I;
        printf("casin(i * (1+2i))/i = %f%+fi\n", creal(z4), cimag(z4));
    }

Output:

    casinh(+0+2i) = 1.316958+1.570796i
    casinh(-0+2i) (the other side of the cut) = -1.316958+1.570796i
    casinh(1+2i) = 1.469352+1.063440i
    casin(i * (1+2i))/i =  1.469352+1.063440i

### References

- C11 standard (ISO/IEC 9899:2011):

  - 7.3.6.2 The casinh functions (p: 192-193)

  - 7.25 Type-generic math <tgmath.h> (p: 373-375)

  - G.6.2.2 The casinh functions (p: 540)

  - G.7 Type-generic math <tgmath.h> (p: 545)

- C99 standard (ISO/IEC 9899:1999):

  - 7.3.6.2 The casinh functions (p: 174-175)

  - 7.22 Type-generic math <tgmath.h> (p: 335-337)

  - G.6.2.2 The casinh functions (p: 475)

  - G.7 Type-generic math <tgmath.h> (p: 480)

### See also

cacoshcacoshfcacoshl (C99)(C99)(C99) | computes the complex arc hyperbolic cosine (function)
catanhcatanhfcatanhl (C99)(C99)(C99) | computes the complex arc hyperbolic tangent (function)
csinhcsinhfcsinhl (C99)(C99)(C99) | computes the complex hyperbolic sine (function)
asinhasinhfasinhl (C99)(C99)(C99) | computes inverse hyperbolic sine (\({\small\operatorname{arsinh}{x} }\)arsinh(x)) (function)
C++ documentation for asinh

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